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Variations for Some Painlevé Equations
Primitivo B. Acosta-Humánezab, Marius van der Putc, Jaap Topc a Instituto Superior de Formación Docente Salomé Ureña – ISFODOSU, Santiago de los Caballeros, Dominican Republic
b School of Basic and Biomedical Sciences, Universidad Simón Bolívar, Barranquilla, Colombia
c Bernoulli Institute, University of Groningen, Groningen, The Netherlands
Аннотация:
This paper first discusses irreducibility of a Painlevé equation $P$. We explain how the Painlevé property is helpful for the computation of special classical and algebraic solutions. As in a paper of Morales-Ruiz we associate an autonomous Hamiltonian $\mathbb{H}$ to a Painlevé equation $P$. Complete integrability of $\mathbb{H}$ is shown to imply that all solutions to $P$ are classical (which includes algebraic), so in particular $P$ is solvable by “quadratures”. Next, we show that the variational equation of $P$ at a given algebraic solution coincides with the normal variational equation of $\mathbb{H}$ at the corresponding solution. Finally, we test the Morales-Ramis theorem in all cases $P_{2}$ to $P_{5}$ where algebraic solutions are present, by showing how our results lead to a quick computation of the component of the identity of the differential Galois group for the first two variational equations. As expected there are no cases where this group is commutative.
Ключевые слова:
Hamiltonian systems, variational equations, Painlevé equations, differential Galois groups.
Поступила: 1 ноября 2018 г.; в окончательном варианте 5 ноября 2019 г.; опубликована 9 ноября 2019 г.
Образец цитирования:
Primitivo B. Acosta-Humánez, Marius van der Put, Jaap Top, “Variations for Some Painlevé Equations”, SIGMA, 15 (2019), 088, 10 pp.
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/sigma1524 https://www.mathnet.ru/rus/sigma/v15/p88
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